In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…
We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of GL+(2,R) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…
Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.
problem Existence and uniqueness of solutions to a generalized Monge-Ampère equation.
method Analyzes the equation's dependence on almost Kähler structure and proves Donaldson's conjecture.
result Proves Donaldson's conjecture for tamed almost complex 4-manifolds.
Infinite genus surfaces have diverse Veech groups.
problem Characterizing Veech groups of infinite genus surfaces.
method Proving every countable subgroup without contracting elements is a Veech group of a surface with infinitely many topological types.
result There are no restrictions on the topological type of surfaces to realize all possible uncountable Veech groups.
We introduce a new technique for finding CAT(-1) surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is geometrically and topologically tame.
Study of homeomorphisms on infinite type surfaces with a classification theorem.
problem Classifying homeomorphisms on surfaces of infinite type.
method Introduce tame homeomorphisms and prove a Nielsen-Thurston type classification theorem.
result For tame homeomorphisms, surfaces decompose into invariant subsurfaces with canonical decompositions.
Researchers find a method to construct projective structures on a specific surface.
problem Constructing projective structures with given holonomy and tameness conditions.
method Grafting circular triangles determined by a natural framing of the representation.
result All structures satisfying the conditions can be obtained through this method.
We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic 2-form which tames the underlying almost complex structure.
We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and J-compatible symplectic cones over a large class of almost complex manifolds…
Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.
problem Analyzing the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
method Building on previous work, the paper characterizes and analyzes the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
result Proves that any locally CB big mapping class group is CB generated and gives an explicit criterion for determining which big mapping class groups are CB generated.
The paper constructs surfaces with infinite-genus Veech groups from finite subgroups of GL(2, R).
problem Constructing surfaces with specific Veech groups from given subgroups.
method PSV construction and modifications to produce tame translation surfaces.
result Ends of constructed surfaces can be decomposed into a group's ends and a dense subset.
We introduce the new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of …
Study on Lee classes of complex surfaces, proving connectedness and bounds.
problem Understanding Lee classes of complex surfaces with LCS structures.
method Analyzing deRham classes of Lee 1-forms and using properties of PSH functions.
result Connectedness of Lee deRham classes and explicit negative upper bound on hyperbolic Kato surfaces.
This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that π_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. Thi…
We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and an E-line is an E-curve of degree 1. We prove that the space of E-lines is agai…
Paper characterizes tamed and weakened tamed four-manifolds using a new technique.
problem Characterizing tamed and weakened tamed four-manifolds.
method Using weakly \widetilde{\mathcal{D}}^+_J -closed technique.
result Characterization of tamed and weakened tamed four-manifolds.
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ:R×X→X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X. Any compact tame set admits plenty tame flows. We prove …
Deep learning models viewed through tame geometry for convergence guarantees.
problem Understanding convergence guarantees in deep learning models.
method Introducing tame geometry concepts and tools for nonsmooth nonconvex settings.
result Illustrates tame geometry as a natural framework for AI systems, especially deep learning.
The study examines when mapping class groups are quasi-isometric to graphs of curves.
problem When is the mapping class group of an infinite-type surface quasi-isometric to a graph of curves?
method Using the work of Rosendal, Mann, and Rafi, the study defines a necessary and sufficient condition called translatability for a mapping class group to be quasi-isometric to a graph of curves.
result The mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard.
We present a short elementary proof of an existence theorem of certain CAT(-1)-surfaces in open hyperbolic 3-manifolds. The main construction lemma in Calegari and Gabai's proof of Marden's Tameness Conjecture can be replaced by an applicable version of our theorem.
Study on convex surfaces in Minkowski space, proving completeness and incompleteness conditions.
problem Characterizing isometric embeddings of hyperbolic plane in Minkowski 3-space.
method Analysis of null support function and curvature conditions.
result Conditions for completeness and incompleteness of convex surfaces.
Paper proposes a new method to stabilize noisy gradient algorithms.
problem Stochastic-gradient Langevin algorithms can introduce bias when taming denominators depend on stochastic-gradient realizations.
method Proposes a structure-preserving framework for designing tamed denominators that avoid unnecessary taming and maintain the stabilizing effect of taming.
result The method avoids stationary bias and explains the stationary error split into bias and remaining error.
Paper proves how to deform circle patterns with interstices.
problem Deforming circle patterns with interstices.
method Combining combinatorial type, exterior angles, and conformal structures.
result Complete description of circle patterns with interstices.
Uncountably many fibrations found on genus 2 handlebody.
problem Finding fibrations on specific 3-manifolds.
method Constructing fibrations with Cantor tree fibers.
result Uncountably many fibrations with non-conjugate monodromies.
The study confirms Gromov's speculation and provides bounds for taming symplectic structures.
problem Understanding the relationship between taming symplectic structures and the area of pseudoholomorphic curves.
method Analyzes the numerical cone of taming symplectic structures and characterizes coarsely holomorphic curves.
result An almost complex manifold with an area bound admits a taming symplectic structure, confirming Gromov's speculation.
Acylindrical hyperbolicity proven for a specific group of automorphisms.
problem Proving the acylindrical hyperbolicity of a group of automorphisms.
method Study of a 2D simplicial complex and proving its contractibility and Gromov-hyperbolicity; finding loxodromic elements.
result Tame automorphism group is acylindrically hyperbolic.
Authors compute Morse-Novikov cohomology for Inoue surfaces and prove nonexistence of certain metrics.
problem Computing and classifying locally conformally Kähler metrics on complex surfaces.
method Review and computation of Morse-Novikov cohomology for known surfaces.
result Nonexistence of LCK metrics with potential on Inoue surfaces and Oeljeklaus-Toma manifolds.
Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the ci…
We show that every non-trivial tame knot or link in R^3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface in R^3 which is a knotted torus must have degree at least eight.
Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.
problem Understanding symplectic embeddings of balls into complex projective spaces, tori, and K3 surfaces.
method Analyzing embeddings with respect to complex structures compatible with the symplectic form and identifying obstructions.
result Symplectic volume is the primary obstruction for the existence of embeddings of balls into certain manifolds.
RELTA-SGLD stabilizes nonconvex SGLD updates with a lighter taming scheme.
problem Stabilizing superlinear stochastic-gradient updates in nonconvex optimization.
method Threshold-based taming with relative-growth principle for stability.
result Polynomial moment stability and first-order stationary accuracy in nonconvex SGLD.
The paper proves symplectic forms on tamed almost complex 4-manifolds.
problem Finding symplectic forms on tamed almost complex 4-manifolds.
method Unified proof approach similar to Buchdahl's Kodaira conjecture proof.
result Existence of new symplectic forms compatible with given almost complex structure.
Study sub-Riemannian surfaces in contact manifolds, proving a Gauss-Bonnet theorem.
problem Proving a Gauss-Bonnet theorem for sub-Riemannian surfaces in contact manifolds.
method Using a family of taming Riemannian metrics, the theorem is derived in the limit.
result Recover topological information of surfaces from geometry around characteristic set.
A closed discrete subset A⊂C is called tame if C∖A is quasiconformally equivalent to C∖Z. By giving several criteria for A to be tame, we shall show that Z+iZ is not tame.
Positivity of intersections in 4-manifolds leads to taming symplectic structures.
problem Taming symplectic structures in almost complex 4-manifolds.
method Proof of positivity of intersections of pseudoholomorphic curves.
result Positivity of intersections is stable and leads to taming symplectic structures.
Whitehead doubles have matching meridional rank and bridge number.
problem Determining the meridional rank of Whitehead doubles.
method Analyzing algebraically tame knots and their Whitehead doubles.
result Meridional rank and bridge number coincide for Whitehead doubles of prime knots.
Global diffeomorphism proof between tame Fréchet spaces.
problem Existence of global diffeomorphism between Fréchet spaces.
method Mountain Pass Theorem and sufficient conditions for diffeomorphism.
result Existence of global diffeomorphism between tame Fréchet spaces.
Every countable compact subset of sphere is tame.
problem Characterizing compact subsets of spheres.
method Proving homeomorphic complements imply homeomorphic subsets.
result Wild subspaces like Antoine contain Cantor sets.
Four-dimensional, oriented Lie algebras g which satisfy the tame-compatible question of Donaldson for all almost complex structures J on g are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are…
We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity…
Develops calculus for tamed Dirichlet spaces using measure theory.
problem Defines calculus for measure spaces with Dirichlet forms.
method Introduces first and second order calculus on tamed Dirichlet spaces.
result Defines various geometric objects on tamed Dirichlet spaces.
This paper shows how path spaces on two-level manifolds can be Hilbert manifold structures.
problem Addressing the structure of path spaces on two-level manifolds.
method Introducing the notion of tameness and constructing charts on path spaces of two-level manifolds.
result Path spaces on tame two-level manifolds have the structure of a Hilbert manifold.
Uniformizing maps and period maps are topologically tame, leading to algebraicity of Hodge loci.
problem Understanding the algebraicity of Hodge loci in arithmetic quotients.
method Proving topological tameness of uniformizing maps and period maps, applying Peterzil-Starchenko's o-minimal GAGA theorem.
result The Hodge locus of (S,V) is a countable union of algebraic subvarieties of S. Study large-scale geometry of infinite type surface mapping class groups.
problem Classify surfaces based on mapping class group properties.
method Coarse geometry, using Rosendal's framework.
result Classification of surfaces based on group properties.
Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form ∂∂ˉ-closed, i.e. to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed …
New algorithm tames non-linear growth in stochastic optimization.
problem Computational challenges in E-step of EM framework.
method Employing interacting particle systems and taming techniques to create tIPLA.
result Non-asymptotic convergence error estimates in Wasserstein-2 distance for tIPLA.
We study the existence of strong Kähler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures J on solvmanifolds G/Γ providing some negative results for some classes of solvmanifolds. In particular, we show that if either J is invariant under the action of a nilpotent complement o…
Classifies pure mapping class groups based on surface properties.
problem Classifying pure mapping class groups based on surface properties.
method Complete classification without additional assumptions.
result PMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface.